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Theorem ltexpri 7269
 Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
Assertion
Ref Expression
ltexpri (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexpri
Dummy variables 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . . . . 8 ((𝑦 = 𝑢𝑧 = 𝑣) → 𝑧 = 𝑣)
21eleq1d 2163 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝑧 ∈ (2nd𝐴) ↔ 𝑣 ∈ (2nd𝐴)))
3 simpl 108 . . . . . . . . 9 ((𝑦 = 𝑢𝑧 = 𝑣) → 𝑦 = 𝑢)
41, 3oveq12d 5708 . . . . . . . 8 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝑧 +Q 𝑦) = (𝑣 +Q 𝑢))
54eleq1d 2163 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 +Q 𝑦) ∈ (1st𝐵) ↔ (𝑣 +Q 𝑢) ∈ (1st𝐵)))
62, 5anbi12d 458 . . . . . 6 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵)) ↔ (𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))))
76cbvexdva 1859 . . . . 5 (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵)) ↔ ∃𝑣(𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))))
87cbvrabv 2632 . . . 4 {𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))} = {𝑢Q ∣ ∃𝑣(𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))}
91eleq1d 2163 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝑧 ∈ (1st𝐴) ↔ 𝑣 ∈ (1st𝐴)))
104eleq1d 2163 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 +Q 𝑦) ∈ (2nd𝐵) ↔ (𝑣 +Q 𝑢) ∈ (2nd𝐵)))
119, 10anbi12d 458 . . . . . 6 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵)) ↔ (𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))))
1211cbvexdva 1859 . . . . 5 (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵)) ↔ ∃𝑣(𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))))
1312cbvrabv 2632 . . . 4 {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))} = {𝑢Q ∣ ∃𝑣(𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))}
148, 13opeq12i 3649 . . 3 ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ = ⟨{𝑢Q ∣ ∃𝑣(𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))}, {𝑢Q ∣ ∃𝑣(𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))}⟩
1514ltexprlempr 7264 . 2 (𝐴<P 𝐵 → ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ ∈ P)
1614ltexprlemfl 7265 . . . 4 (𝐴<P 𝐵 → (1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) ⊆ (1st𝐵))
1714ltexprlemrl 7266 . . . 4 (𝐴<P 𝐵 → (1st𝐵) ⊆ (1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)))
1816, 17eqssd 3056 . . 3 (𝐴<P 𝐵 → (1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (1st𝐵))
1914ltexprlemfu 7267 . . . 4 (𝐴<P 𝐵 → (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) ⊆ (2nd𝐵))
2014ltexprlemru 7268 . . . 4 (𝐴<P 𝐵 → (2nd𝐵) ⊆ (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)))
2119, 20eqssd 3056 . . 3 (𝐴<P 𝐵 → (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (2nd𝐵))
22 ltrelpr 7161 . . . . . . 7 <P ⊆ (P × P)
2322brel 4519 . . . . . 6 (𝐴<P 𝐵 → (𝐴P𝐵P))
2423simpld 111 . . . . 5 (𝐴<P 𝐵𝐴P)
25 addclpr 7193 . . . . 5 ((𝐴P ∧ ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ ∈ P) → (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) ∈ P)
2624, 15, 25syl2anc 404 . . . 4 (𝐴<P 𝐵 → (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) ∈ P)
2723simprd 113 . . . 4 (𝐴<P 𝐵𝐵P)
28 preqlu 7128 . . . 4 (((𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) ∈ P𝐵P) → ((𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵 ↔ ((1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (1st𝐵) ∧ (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (2nd𝐵))))
2926, 27, 28syl2anc 404 . . 3 (𝐴<P 𝐵 → ((𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵 ↔ ((1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (1st𝐵) ∧ (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (2nd𝐵))))
3018, 21, 29mpbir2and 893 . 2 (𝐴<P 𝐵 → (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵)
31 oveq2 5698 . . . 4 (𝑥 = ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ → (𝐴 +P 𝑥) = (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩))
3231eqeq1d 2103 . . 3 (𝑥 = ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ → ((𝐴 +P 𝑥) = 𝐵 ↔ (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵))
3332rspcev 2736 . 2 ((⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ ∈ P ∧ (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵) → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
3415, 30, 33syl2anc 404 1 (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ∃wrex 2371  {crab 2374  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690  1st c1st 5947  2nd c2nd 5948  Qcnq 6936   +Q cplq 6938  Pcnp 6947   +P cpp 6949
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